The invention provides for the measurement of solute fluxes and viscous flow in liquid and partially saturated porous media. The solutes may be organic or inorganic molecules that are dissolved and/or attached to dispersed colloids in the liquid. More specifically, the invention relates to the monitoring of solute fluxes in water resources, including, but not limited to, variably saturated soils, sediments and groundwater aquifers, surface water, water for industrial purposes, tap water, drinking water and aqueous waste streams.
Monitoring for solutes in such water resources is often needed to verify concentrations of harmful substances in relation to specific environmental standards or reference levels as proclaimed by the regulating authorities. So, the solutes that are of interest for the present invention represents a very diverse group of organic and inorganic compounds. Depending on the specific environment, they may include for example petroleum or tar-based compounds, halogenated solvents, heavy metals, macronutrients, radionuclides, biocides and their metabolites, surfactants, hormones, pharmaceutical products and their metabolites etc.
The displacement of solute mass in variously saturated porous media occurs by the combination of convection, Jm, diffusion, JD, and hydrodynamic dispersion, Jh (Van Genuchten en Wierenga, 1986). The convective transport is the displacement of the solute along with the viscous flow of the liquid, and is described by the equation:Jm=qC  (1)where Jm is the convective transport flux (g cm−2 s−1), q is the volumetric flux density of a viscous fluid (cm3 cm−2 s−1) and C is the solute concentration (g cm−3).
Solute diffusion results from random Brownian motion of molecules in solutions and in variously saturated media. When the concentration gradient and the volumetric fluid content, θ, are constant, diffusion of a non-sorbing solute may be described by Fick's first law:
                              J          D                =                              -            θ                    ⁢                                          ⁢                      D            m                    ⁢                                    ∂              C                                      ∂              x                                                          (        2        )            
where JD is the diffusive transport flux (g cm−2 s−1), θ is the volumetric liquid content of the porous medium (cm3 cm−3), Dm is the diffusion coefficient of the porous medium (cm2 s−1), and x is the space coordinate. Diffusion in a semi-infinite solution or porous medium does not move the centre of mass of a solute, as the Brownian forces move the molecules away from the centre of mass in all directions. However, in a finite medium with heterogeneous boundaries, diffusion moves the centre of mass away from a source zone and towards a sink zone along the concentration gradient in the medium.
The diffusion coefficient Dm in porous media is always smaller than the diffusion coefficient of the molecule in a free bulk liquid, Do. This is due to the tortuous pathway of the connecting pores, and, among other things, van der Waals interactions with the solid surface. The two diffusion coefficients Dm and Do are linearly related:Dm=kθ(L/Le)2Do  (3)where k is an empirical constant, L and Le are the straight distance between two points and the pathway laid out by the pore system, respectively. Hence, the quadratic term in equation 3 accounts for the pore tortuosity. The relationship between Dm and θ is highly non-linear, because the tortuosity increases with decreasing water content.
Hydrodynamic dispersion results from pore scale heterogeneity of the pore water velocity magnitude and direction. It has been shown that the dispersion effect may be described mathematically similarly to diffusive transport:
                              J          h                =                              -            θ                    ⁢                                          ⁢                      D            h                    ⁢                                    ∂              C                                      ∂              x                                                          (        4        )            Where Jh is the dispersive transport flux (g cm−2 s−1), and Dh is the mechanical dispersion coefficient (cm2 s−1). The mechanical dispersion coefficient is increasing with increasing fluid velocity according to the empirical relationship:Dh=λν″  (5)Where λ is the dispersivity (cm), v is the pore water velocity (cm s−1), that is approximated as q/θ, and n is an empirical parameter, normally equal to 1 (Van Genuchten en Wierenga, 1986). From this relationship it follows that the dispersive flux, unlike the diffusive flux term, vanishes for v→0.
Combining the three terms contributing to the displacement, an expression for the total solute flux, Js, is obtained:
                              J          s                =                                            -              θ                        ⁢                                                  ⁢                          (                                                D                  h                                +                                  D                  m                                            )                        ⁢                                          ∂                C                                            ∂                x                                              +          qC                                    (        6        )            Substitution of this equation into the equation of continuity for a solute that does not undergo irreversible reactions:
                                          ∂                          ∂              t                                ⁢                      (                                          θ                ⁢                                                                  ⁢                C                            +                              ρ                ⁢                                                                  ⁢                S                                      )                          =                  -                                    ∂                              J                s                                                    ∂              x                                                          (        7        )            Yields the general transport equation:
                                          ∂                          ∂              t                                ⁢                      (                                          θ                ⁢                                                                  ⁢                C                            +                              ρ                ⁢                                                                  ⁢                S                                      )                          =                              -                          ∂                              ∂                x                                              ⁢                      (                                                            θ                  ⁡                                      (                                                                  D                        m                                            +                                              D                                                  h                          ⁢                                                                                                                                                                      )                                                  ⁢                                                      ∂                    C                                                        ∂                    x                                                              -              qC                        )                                              (        8        )            Where t is time (s), ρ is the bulk density of the porous medium (g cm−3), and S is the amount of solute adsorbed to the solute phase (g g−1). The simplest form in which sorption to the solid phase can be represented is by instantaneous linear equilibrium conditions:S=KdC  (9)Where Kd is the slope of the linear isotherm (cm3 g−1). When considering steady liquid flow in a homogenous porous medium, implying that θ and q are constant in space and time, the transport equation reduces to:
                                          ∂            C                                ∂            t                          =                                                            (                                                      D                    h                                    +                                      D                    m                                                  )                            R                        ⁢                                                            ∂                  2                                ⁢                C                                            ∂                                  x                  2                                                              -                                    v              R                        ⁢                                          ∂                C                                            ∂                x                                                                        (        10        )            Where R is the retardation factor:
                    R        =                  1          +                                    ρ              ⁢                                                          ⁢                              K                d                                      θ                                              (        11        )            Equation 8 may be solved numerically for transient conditions, that is, θ and q varies both in time and space, and dynamic boundary conditions. Numerical solution codes are available both in one and two dimensions. Equation 10 may be solved analytically for constant boundary conditions (e.g. Wierenga and Van Genuchten, 1986).
Gaseous transport in unsaturated porous media occurs due to diffusion processes only, unless pressure gradients are present. The diffusion process may again be described by Fick's first law:
                              J          D                =                              -            θ                    ⁢                                          ⁢                      D            g                    ⁢                                    ∂                              C                g                                                    ∂              x                                                          (        12        )            Where Dg is the gaseous diffusion coefficient and Cg is the partial gas concentration in the gas filled pores. Analogue to solute diffusion, the gas diffusion is highly non-linear with respect to the volume contributing to the displacement, i.e. air-filled porosity. Hence, it follows that at a high liquid filled porosity, liquid displacement and solute diffusion dominate the transport process, while at high air-filled porosity, gas-diffusion will dominate the displacement of a volatile compound in the porous medium.